A129639 -id:A129639 - cp's OEIS frontend

A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 4, 2, 0, 0, 1, 5, 6, 6, 2, 0, 0, 1, 6, 8, 10, 8, 2, 0, 0, 1, 7, 10, 14, 16, 12, 2, 0, 0, 1, 8, 12, 18, 24, 26, 16, 2, 0, 0, 1, 9, 14, 22, 32, 42, 42, 24, 2, 0, 0, 1, 10, 16, 26, 40, 58, 72, 68, 32, 2, 0, 0

Offset: 0

Alois P. Heinz, Dec 03 2012

Equivalently, the number of walks of length n-1 on the path graph P_k. - Andrew Howroyd, Apr 17 2017

			A(5,3) = 12: there are 12 words of length 5 over 3-ary alphabet {a,b,c}, where neighboring letters are neighbors in the alphabet: ababa, ababc, abcba, abcbc, babab, babcb, bcbab, bcbcb, cbaba, cbabc, cbcba, cbcbc.
Square array A(n,k) begins:
  1,  1,  1,  1,  1,   1,   1,   1, ...
  0,  1,  2,  3,  4,   5,   6,   7, ...
  0,  0,  2,  4,  6,   8,  10,  12, ...
  0,  0,  2,  6, 10,  14,  18,  22, ...
  0,  0,  2,  8, 16,  24,  32,  40, ...
  0,  0,  2, 12, 26,  42,  58,  74, ...
  0,  0,  2, 16, 42,  72, 104, 136, ...
  0,  0,  2, 24, 68, 126, 188, 252, ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0, 2-10 give: A000007, A040000, A029744(n+2) for n>0, A006355(n+3) for n>0, A090993(n+1) for n>0, A090995(n-1) for n>2, A129639, A153340, A153362, A153360.
Rows 0-6 give: A000012, A001477, A005843(k-1) for k>0, A016825(k-2) for k>1, A008590(k-2) for k>2, A113770(k-2) for k>3, A063164(k-2) for k>4.
Main diagonal gives: A102699.
Cf. A198632, A188866, A276562, A208727, A208671.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0, add(b(n-1, j, k), j=1..k),
      `if`(i>1, b(n-1, i-1, k), 0)+
      `if`(i b(n, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, Sum[b[n-1, j, k], {j, 1, k}], If[i>1, b[n-1, i-1, k], 0] + If[iJean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

TransferGf(m,u,t,v,z)=vector(m,i,u(i))*matsolve(matid(m)-z*matrix(m,m,i,j,t(i,j)),vectorv(m,i,v(i)));
ColGf(m,z)=1+z*TransferGf(m, i->1, (i,j)->abs(i-j)==1, j->1, z);
a(n,k)=Vec(ColGf(k,x) + O(x^(n+1)))[n+1];
for(n=0, 7, for(k=0, 7, print1( a(n,k), ", ") ); print(); );
\\ Andrew Howroyd, Apr 17 2017

A116183 Array T(k,n) = number of meaningful differential operations of the n-th order on the space R^(3+k), for k=>0, n>0, read by antidiagonals.

Original entry on oeis.org

3, 4, 5, 5, 6, 8, 6, 9, 8, 13, 7, 10, 16, 12, 21, 8, 13, 16, 29, 16, 34, 9, 14, 24, 26, 52, 24, 55, 10, 17, 24, 45, 42, 94, 32, 89, 11, 18, 32, 42, 84, 68, 169

Offset: 1

Jonathan Vos Post, Apr 08 2007

Two more rows can be obtained from A129638 and A129639.

			Table begins:
k=0.|.3..5..8.13..21..34..55..89..144..233..377..610..987.1597...
k=1.|.4..6..8.12..16..24..32..48...64...96..128..192..256..384...
k=2.|.5..9.16.29..52..94.169.305..549..990.1783.3214.5790...
k=3.|.6.10.16.26..42..68.110.178..288..466..754.1220.1974...
k=4.|.7.13.24.45..84.158.296.557.1045.1966.3691.6942.13038...
k=5.|.8.14.24.42..72.126.216.378..648.1134.1944.3402..5832...
k=6.|.9.17.32.61.116.222.424.813.1556.2986.5721.10982...
k=7.|10.18.32.58.104.188.338.610.1098.1980.3566.6428...

Branko Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33; arXiv:0704.0750 [math.DG], 2007.

k=0 row is A020701. k=1 row is A090989. k=2 row is A090990. k=3 row is A090991. k=4 row is A090992. k=5 row is A090993. k=6 row is A090994. k=7 row is A090995.

Diagonal: A127935.

A127935 Number of meaningful differential operations of the n-th order on the space R^(2+n).

Original entry on oeis.org

3, 6, 16, 26, 84, 126, 424, 610, 2068, 2936, 9816, 13884, 45608, 64750, 208336, 297570, 938676, 1351492, 4181752, 6071028, 18454648, 27023598, 80796336, 119300636, 351331464, 522981328, 1518742384, 2278188504, 6531607248, 9869753934, 27963677600, 42547990626

Offset: 1

Jonathan Vos Post, Apr 09 2007, Jun 08 2007

nonn

			a(1) = 3 = A020701(1) is number of meaningful differential operations of the first order on the space R^3, namely {div, grad, curl}.
a(2) = 6 = A090989(2) is number of meaningful differential operations of the 2nd order on the space R^4 (some of them are identically zero though).
a(3) = 16 = A090990(3) is number of meaningful differential operations of the 3rd order on the space R^5.

R. Bott, L. W. Tu, Differential forms in algebraic topology, New York: Springer, 1982.

Branko Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33; arXiv:0704.0750 [math.DG], 2007.

Main diagonal of A116183.

Cf. A020701, A090989-A090995, A129638, A129639.

r[n_] := Table[Boole[j == i + 1 || i + j == n + 1], {i, n}, {j, n}];
Table[Total@Total@If[n == 1, IdentityMatrix[3], MatrixPower[r[n+2], n-1]], {n, 10}]
(* Andrey Zabolotskiy, Apr 30 2021 *)

Corrected from 8th term onwards. It appears the 8th and 9th terms listed were incorrectly taken from A000045 with two numbers concatenated together, whereas the 8th, 9th and 10th terms should have been the 8th term of A090995, the 9th of A129638 and the 10th of A129639. Joseph Myers, Dec 23 2008

Name and examples corrected, terms a(11) and beyond added by Andrey Zabolotskiy, Apr 30 2021

A208670 Number of 2n-bead necklaces labeled with numbers 1..7 allowing reversal, with neighbors differing by exactly 1.

Original entry on oeis.org

6, 11, 20, 41, 86, 208, 514, 1398, 3934, 11576, 34850, 107303, 334396, 1053851, 3345320, 10685570, 34292064, 110498897, 357256018, 1158492478, 3766458404, 12274037845, 40082339406, 131144365904, 429838330172, 1411104048106, 4639351259122, 15273992343065

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

			All solutions for n=3:
..1....5....1....5....3....4....1....6....4....1....2....2....4....5....3....4
..2....6....2....6....4....5....2....7....5....2....3....3....5....6....4....5
..1....5....3....7....3....4....1....6....6....3....4....2....6....5....5....4
..2....6....2....6....4....5....2....7....5....4....3....3....7....6....4....5
..3....5....3....7....5....6....1....6....6....3....4....2....6....7....5....4
..2....6....2....6....4....5....2....7....5....2....3....3....5....6....4....5
..
..2....2....3....3
..3....3....4....4
..2....4....3....5
..3....5....4....6
..4....4....3....5
..3....3....4....4

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 7 of A208671.

a(n) = (2*A208726(n) + A129639(11+n))/4.

a(12)-a(28) from Andrew Howroyd, Mar 19 2017

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A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A116183 Array T(k,n) = number of meaningful differential operations of the n-th order on the space R^(3+k), for k=>0, n>0, read by antidiagonals.

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A127935 Number of meaningful differential operations of the n-th order on the space R^(2+n).

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A208670 Number of 2n-bead necklaces labeled with numbers 1..7 allowing reversal, with neighbors differing by exactly 1.

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